Hoare’s vs Lomuto partition scheme in QuickSort

We have discussed the implementation of QuickSort using Lomuto partition scheme. Lomuto’s partition scheme is easy to implement as compared to Hoare scheme. This has inferior performance to Hoare’s QuickSort.

Lomuto’s Partition Scheme:

This algorithm works by assuming the pivot element as the last element. If any other element is given as a pivot element then swap it first with the last element. Now initialize two variables i as low and j also low,  iterate over the array and increment i when arr[j] <= pivot and swap arr[i] with arr[j] otherwise increment only j. After coming out from the loop swap arr[i] with arr[hi]. This i stores the pivot element.

partition(arr[], lo, hi) 
    pivot = arr[hi]
    i = lo-1     // place for swapping
    for j := lo to hi – 1 do
        if arr[j] <= pivot then
            i = i + 1 
            swap arr[i] with arr[j]
    swap arr[i+1] with arr[hi]
    return i+1

Refer QuickSort for details of this partitioning scheme. 
Below are implementations of this approach:-

C++

/* C++ implementation QuickSort using Lomuto's partition
   Scheme.*/
#include<bits/stdc++.h>
using namespace std;

/* This function takes last element as pivot, places
   the pivot element at its correct position in sorted
    array, and places all smaller (smaller than pivot)
   to left of pivot and all greater elements to right
   of pivot */
int partition(int arr[], int low, int high)
{
    int pivot = arr[high];    // pivot
    int i = (low - 1);  // Index of smaller element

    for (int j = low; j <= high- 1; j++)
    {
        // If current element is smaller than or
        // equal to pivot
        if (arr[j] <= pivot)
        {
            i++;    // increment index of smaller element
            swap(arr[i], arr[j]);
        }
    }
    swap(arr[i + 1], arr[high]);
    return (i + 1);
}

/* The main function that implements QuickSort
 arr[] --> Array to be sorted,
  low  --> Starting index,
  high  --> Ending index */
void quickSort(int arr[], int low, int high)
{
    if (low < high)
    {
        /* pi is partitioning index, arr[p] is now
           at right place */
        int pi = partition(arr, low, high);

        // Separately sort elements before
        // partition and after partition
        quickSort(arr, low, pi - 1);
        quickSort(arr, pi + 1, high);
    }
}

/* Function to print an array */
void printArray(int arr[], int size)
{
    int i;
    for (i=0; i < size; i++)
        printf("%d ", arr[i]);
    printf("\n");
}

// Driver program to test above functions
int main()
{
    int arr[] = {10, 7, 8, 9, 1, 5};
    int n = sizeof(arr)/sizeof(arr[0]);
    quickSort(arr, 0, n-1);
    printf("Sorted array: \n");
    printArray(arr, n);
    return 0;
}

Java

// Java implementation QuickSort
// using Lomuto's partition Scheme
import java.io.*;

class GFG
{
static void Swap(int[] array,
                 int position1,
                 int position2)
{
    // Swaps elements in an array

    // Copy the first position's element
    int temp = array[position1];

    // Assign to the second element
    array[position1] = array[position2];

    // Assign to the first element
    array[position2] = temp;
}

/* This function takes last element as
pivot, places the pivot element at its
correct position in sorted array, and
places all smaller (smaller than pivot)
to left of pivot and all greater elements
to right of pivot */
static int partition(int []arr, int low,
                                int high)
{
    int pivot = arr[high];

    // Index of smaller element
    int i = (low - 1);

    for (int j = low; j <= high- 1; j++)
    {
        // If current element is smaller
        // than or equal to pivot
        if (arr[j] <= pivot)
        {
            i++; // increment index of
                 // smaller element
            Swap(arr, i, j);
        }
    }
    Swap(arr, i + 1, high);
    return (i + 1);
}

/* The main function that
   implements QuickSort
arr[] --> Array to be sorted,
low --> Starting index,
high --> Ending index */
static void quickSort(int []arr, int low,
                                 int high)
{
    if (low < high)
    {
        /* pi is partitioning index,
        arr[p] is now at right place */
        int pi = partition(arr, low, high);

        // Separately sort elements before
        // partition and after partition
        quickSort(arr, low, pi - 1);
        quickSort(arr, pi + 1, high);
    }
}

/* Function to print an array */
static void printArray(int []arr, int size)
{
    int i;
    for (i = 0; i < size; i++)
    System.out.print(" " + arr[i]);
    System.out.println();
}

// Driver Code
static public void main (String[] args)
{
    int []arr = {10, 7, 8, 9, 1, 5};
    int n = arr.length;
    quickSort(arr, 0, n-1);
    System.out.println("Sorted array: ");
    printArray(arr, n);
}
}

// This code is contributed by vt_m.

Python3

''' Python3 implementation QuickSort using Lomuto's partition
Scheme.'''

''' This function takes last element as pivot, places
the pivot element at its correct position in sorted
    array, and places all smaller (smaller than pivot)
to left of pivot and all greater elements to right
of pivot '''
def partition(arr, low, high):

    # pivot
    pivot = arr[high]

    # Index of smaller element
    i = (low - 1)
    for j in range(low, high):

        # If current element is smaller than or
        # equal to pivot
        if (arr[j] <= pivot):

            # increment index of smaller element
            i += 1
            arr[i], arr[j] = arr[j], arr[i]
    arr[i + 1], arr[high] = arr[high], arr[i + 1]
    return (i + 1)

''' The main function that implements QuickSort
arr --> Array to be sorted,
low --> Starting index,
high --> Ending index '''
def quickSort(arr, low, high):
    if (low < high):

        ''' pi is partitioning index, arr[p] is now
        at right place '''
        pi = partition(arr, low, high)

        # Separately sort elements before
        # partition and after partition
        quickSort(arr, low, pi - 1)
        quickSort(arr, pi + 1, high)

''' Function to print an array '''
def printArray(arr, size):

    for i in range(size):
        print(arr[i], end = " ")
    print()

# Driver code

arr = [10, 7, 8, 9, 1, 5]
n = len(arr)
quickSort(arr, 0, n - 1)
print("Sorted array:")
printArray(arr, n)

# This code is contributed by SHUBHAMSINGH10

C#

// C# implementation QuickSort
// using Lomuto's partition Scheme
using System;

class GFG
{
static void Swap(int[] array,
                 int position1,
                 int position2)
{
    // Swaps elements in an array

    // Copy the first position's element
    int temp = array[position1];

    // Assign to the second element
    array[position1] = array[position2];

    // Assign to the first element
    array[position2] = temp;
}

/* This function takes last element as
pivot, places the pivot element at its
correct position in sorted array, and
places all smaller (smaller than pivot)
to left of pivot and all greater elements
to right of pivot */
static int partition(int []arr, int low,
                                int high)
{
    int pivot = arr[high];

    // Index of smaller element
    int i = (low - 1);

    for (int j = low; j <= high- 1; j++)
    {
        // If current element is smaller
        // than or equal to pivot
        if (arr[j] <= pivot)
        {
            i++; // increment index of
                 // smaller element
            Swap(arr, i, j);
        }
    }
    Swap(arr, i + 1, high);
    return (i + 1);
}

/* The main function that
   implements QuickSort
arr[] --> Array to be sorted,
low --> Starting index,
high --> Ending index */
static void quickSort(int []arr, int low,
                                 int high)
{
    if (low < high)
    {
        /* pi is partitioning index,
        arr[p] is now at right place */
        int pi = partition(arr, low, high);

        // Separately sort elements before
        // partition and after partition
        quickSort(arr, low, pi - 1);
        quickSort(arr, pi + 1, high);
    }
}

/* Function to print an array */
static void printArray(int []arr, int size)
{
    int i;
    for (i = 0; i < size; i++)
    Console.Write(" " + arr[i]);
    Console.WriteLine();
}

// Driver Code
static public void Main()
{
    int []arr = {10, 7, 8, 9, 1, 5};
    int n = arr.Length;
    quickSort(arr, 0, n-1);
    Console.WriteLine("Sorted array: ");
    printArray(arr, n);
}
}

// This code is contributed by vt_m.

Javascript

<script>

// JavaScript implementation QuickSort
// using Lomuto's partition Scheme

function Swap(array, position1, position2)
{
    // Swaps elements in an array

    // Copy the first position's element
    let temp = array[position1];

    // Assign to the second element
    array[position1] = array[position2];

    // Assign to the first element
    array[position2] = temp;
}

/* This function takes last element as
pivot, places the pivot element at its
correct position in sorted array, and
places all smaller (smaller than pivot)
to left of pivot and all greater elements
to right of pivot */
function partition(arr, low, high)
{
    let pivot = arr[high];

    // Index of smaller element
    let i = (low - 1);

    for (let j = low; j <= high- 1; j++)
    {
        // If current element is smaller
        // than or equal to pivot
        if (arr[j] <= pivot)
        {
            i++; // increment index of
                 // smaller element
            Swap(arr, i, j);
        }
    }
    Swap(arr, i + 1, high);
    return (i + 1);
}

/* The main function that
   implements QuickSort
arr[] --> Array to be sorted,
low --> Starting index,
high --> Ending index */
function quickSort(arr, low, high)
{
    if (low < high)
    {
        /* pi is partitioning index,
        arr[p] is now at right place */
        let pi = partition(arr, low, high);

        // Separately sort elements before
        // partition and after partition
        quickSort(arr, low, pi - 1);
        quickSort(arr, pi + 1, high);
    }
}

/* Function to print an array */
function printArray(arr, size)
{
    let i;
    for (i = 0; i < size; i++)
    document.write(" " + arr[i]);
    document.write("<br/>");
}

// Driver Code
    let arr = [10, 7, 8, 9, 1, 5];
    let n = arr.length;
    quickSort(arr, 0, n-1);
    document.write("Sorted array: ");
    printArray(arr, n);

 // This code is contributed by chinmoy1997pal.
</script>

Output

Sorted array: 
1 5 7 8 9 10 

Time Complexity: O(N²) 
Auxiliary Space: O(1) 

Hoare’s Partition Scheme:

Hoare’s Partition Scheme works by initializing two indexes that start at two ends, the two indexes move toward each other until an inversion is (A smaller value on the left side and greater value on the right side) found. When an inversion is found, two values are swapped and the process is repeated.

Algorithm:

partition(arr[], lo, hi)
   pivot = arr[lo]
   i = lo - 1  // Initialize left index
   j = hi + 1  // Initialize right index

   // Find a value in left side greater
   // than pivot
   do
      i = i + 1
   while arr[i] < pivot

   // Find a value in right side smaller
   // than pivot
   do
      j--;
   while (arr[j] > pivot);

   if i >= j then 
      return j

   swap arr[i] with arr[j]

Below are implementations of this approach:- 

C++

/* C++ implementation of QuickSort using Hoare's
   partition scheme. */
#include <bits/stdc++.h>
using namespace std;

/* This function takes first element as pivot, and places
   all the elements smaller than the pivot on the left side
   and all the elements greater than the pivot on
   the right side. It returns the index of the last element
   on the smaller side*/
int partition(int arr[], int low, int high)
{
    int pivot = arr[low];
    int i = low - 1, j = high + 1;

    while (true) {
        // Find leftmost element greater than
        // or equal to pivot
        do {
            i++;
        } while (arr[i] < pivot);

        // Find rightmost element smaller than
        // or equal to pivot
        do {
            j--;
        } while (arr[j] > pivot);

        // If two pointers met.
        if (i >= j)
            return j;

        swap(arr[i], arr[j]);
    }
}

/* The main function that implements QuickSort
 arr[] --> Array to be sorted,
  low  --> Starting index,
  high  --> Ending index */
void quickSort(int arr[], int low, int high)
{
    if (low < high) {
        /* pi is partitioning index, arr[p] is now
           at right place */
        int pi = partition(arr, low, high);

        // Separately sort elements before
        // partition and after partition
        quickSort(arr, low, pi);
        quickSort(arr, pi + 1, high);
    }
}

/* Function to print an array */
void printArray(int arr[], int n)
{
    for (int i = 0; i < n; i++)
        printf("%d ", arr[i]);
    printf("\n");
}

// Driver Code
int main()
{
    int arr[] = { 10, 7, 8, 9, 1, 5 };
    int n = sizeof(arr) / sizeof(arr[0]);
    quickSort(arr, 0, n - 1);
    printf("Sorted array: \n");
    printArray(arr, n);
    return 0;
}

Java

// Java implementation of QuickSort
// using Hoare's partition scheme
import java.io.*;

class GFG {

    /* This function takes first element as pivot, and
       places all the elements smaller than the pivot on the
       left side and all the elements greater than the pivot
       on the right side. It returns the index of the last
       element on the smaller side*/
    static int partition(int[] arr, int low, int high)
    {
        int pivot = arr[low];
        int i = low - 1, j = high + 1;

        while (true) {
            // Find leftmost element greater
            // than or equal to pivot
            do {
                i++;
            } while (arr[i] < pivot);

            // Find rightmost element smaller
            // than or equal to pivot
            do {
                j--;
            } while (arr[j] > pivot);

            // If two pointers met.
            if (i >= j)
                return j;
            int temp = arr[i];
            arr[i] = arr[j];
            arr[j] = temp;
            // swap(arr[i], arr[j]);
        }
    }

    /* The main function that
       implements QuickSort
    arr[] --> Array to be sorted,
    low --> Starting index,
    high --> Ending index */
    static void quickSort(int[] arr, int low, int high)
    {
        if (low < high) {
            /* pi is partitioning index,
            arr[p] is now at right place */
            int pi = partition(arr, low, high);

            // Separately sort elements before
            // partition and after partition
            quickSort(arr, low, pi);
            quickSort(arr, pi + 1, high);
        }
    }

    /* Function to print an array */
    static void printArray(int[] arr, int n)
    {
        for (int i = 0; i < n; i++)
            System.out.print(" " + arr[i]);
        System.out.println();
    }

    // Driver Code
    static public void main(String[] args)
    {
        int[] arr = { 10, 7, 8, 9, 1, 5 };
        int n = arr.length;
        quickSort(arr, 0, n - 1);
        System.out.println("Sorted array: ");
        printArray(arr, n);
    }
}

// This code is contributed by vt_m.

Python3

''' Python implementation of QuickSort using Hoare's
partition scheme. '''

''' This function takes first element as pivot, and places
      all the elements smaller than the pivot on the left side
      and all the elements greater than the pivot on
      the right side. It returns the index of the last element
      on the smaller side '''


def partition(arr, low, high):

    pivot = arr[low]
    i = low - 1
    j = high + 1

    while (True):

        # Find leftmost element greater than
        # or equal to pivot
        i += 1
        while (arr[i] < pivot):
            i += 1

        # Find rightmost element smaller than
        # or equal to pivot
        j -= 1
        while (arr[j] > pivot):
            j -= 1

        # If two pointers met.
        if (i >= j):
            return j

        arr[i], arr[j] = arr[j], arr[i]


''' The main function that implements QuickSort
arr --> Array to be sorted,
low --> Starting index,
high --> Ending index '''


def quickSort(arr, low, high):
    ''' pi is partitioning index, arr[p] is now
    at right place '''
    if (low < high):

        pi = partition(arr, low, high)

        # Separately sort elements before
        # partition and after partition
        quickSort(arr, low, pi)
        quickSort(arr, pi + 1, high)


''' Function to print an array '''


def printArray(arr, n):
    for i in range(n):
        print(arr[i], end=" ")
    print()


# Driver code
arr = [10, 7, 8, 9, 1, 5]
n = len(arr)
quickSort(arr, 0, n - 1)
print("Sorted array:")
printArray(arr, n)

# This code is contributed by shubhamsingh10

C#

// C# implementation of QuickSort
// using Hoare's partition scheme
using System;

class GFG {

    /* This function takes first element as pivot, and
       places all the elements smaller than the pivot on the
       left side and all the elements greater than the pivot
       on the right side. It returns the index of the last
       element on the smaller side*/
    static int partition(int[] arr, int low, int high)
    {
        int pivot = arr[low];
        int i = low - 1, j = high + 1;

        while (true) {
            // Find leftmost element greater
            // than or equal to pivot
            do {
                i++;
            } while (arr[i] < pivot);

            // Find rightmost element smaller
            // than or equal to pivot
            do {
                j--;
            } while (arr[j] > pivot);

            // If two pointers met.
            if (i >= j)
                return j;
            int temp = arr[i];
            arr[i] = arr[j];
            arr[j] = temp;
            // swap(arr[i], arr[j]);
        }
    }

    /* The main function that
       implements QuickSort
    arr[] --> Array to be sorted,
    low --> Starting index,
    high --> Ending index */
    static void quickSort(int[] arr, int low, int high)
    {
        if (low < high) {
            /* pi is partitioning index,
            arr[p] is now at right place */
            int pi = partition(arr, low, high);

            // Separately sort elements before
            // partition and after partition
            quickSort(arr, low, pi);
            quickSort(arr, pi + 1, high);
        }
    }

    /* Function to print an array */
    static void printArray(int[] arr, int n)
    {
        for (int i = 0; i < n; i++)
            Console.Write(" " + arr[i]);
        Console.WriteLine();
    }

    // Driver Code
    static public void Main()
    {
        int[] arr = { 10, 7, 8, 9, 1, 5 };
        int n = arr.Length;
        quickSort(arr, 0, n - 1);
        Console.WriteLine("Sorted array: ");
        printArray(arr, n);
    }
}

// This code is contributed by vt_m.

Javascript

<script>
    // Javascript implementation of QuickSort
    // using Hoare's partition scheme

    /* This function takes first element as pivot, and
       places all the elements smaller than the pivot on the
       left side and all the elements greater than the pivot
       on the right side. It returns the index of the last
       element on the smaller side*/
    function partition(arr, low, high)
    {
        let pivot = arr[low];
        let i = low - 1, j = high + 1;

        while (true) {
            // Find leftmost element greater
            // than or equal to pivot
            do {
                i++;
            } while (arr[i] < pivot);

            // Find rightmost element smaller
            // than or equal to pivot
            do {
                j--;
            } while (arr[j] > pivot);

            // If two pointers met.
            if (i >= j)
                return j;
            let temp = arr[i];
            arr[i] = arr[j];
            arr[j] = temp;
            // swap(arr[i], arr[j]);
        }
    }

    /* The main function that
       implements QuickSort
    arr[] --> Array to be sorted,
    low --> Starting index,
    high --> Ending index */
    function quickSort(arr, low, high)
    {
        if (low < high) {
            /* pi is partitioning index,
            arr[p] is now at right place */
            let pi = partition(arr, low, high);

            // Separately sort elements before
            // partition and after partition
            quickSort(arr, low, pi);
            quickSort(arr, pi + 1, high);
        }
    }

    /* Function to print an array */
    function printArray(arr, n)
    {
        for (let i = 0; i < n; i++)
            document.write(" " + arr[i]);
        document.write("</br>");
    }

    let arr = [ 10, 7, 8, 9, 1, 5 ];
    let n = arr.length;
    quickSort(arr, 0, n - 1);
    document.write("Sorted array: " + "</br>");
    printArray(arr, n);

</script>

Output

Sorted array: 
1 5 7 8 9 10 

Time Complexity: O(N) 
Auxiliary Space: O(1)

Note : If we change Hoare’s partition to pick the last element as pivot, then the Hoare’s partition may cause QuickSort to go into an infinite recursion. For example, {10, 5, 6, 20} and pivot is arr[high], then returned index will always be high and call to same QuickSort will be made. To handle a random pivot, we can always swap that random element with the first element and simply follow the above algorithm.
Comparison:

1. Hoare’s scheme is more efficient than Lomuto’s partition scheme because it does three times fewer swaps on average, and it creates efficient partitions even when all values are equal.
2. Like Lomuto’s partition scheme, Hoare partitioning also causes Quick sort to degrade to O(n^2) when the input array is already sorted, it also doesn’t produce a stable sort.
3. Note that in this scheme, the pivot’s final location is not necessarily at the index that was returned, and the next two segments that the main algorithm recurs on are (lo..p) and (p+1..hi) as opposed to (lo..p-1) and (p+1..hi) as in Lomuto’s scheme.
4. Both Hoare’s Partition, as well as Lomuto’s partition, are unstable.

Hoare partition algorithm	Lomuto partition algorithm
Generally, the first item or the element is assumed to be the initial pivot element. Some choose the middle element and even the last element.	Generally, a random element of the array is located and picked and then exchanged with the first or the last element to give initial pivot values. In the aforementioned algorithm, the last element of the list is considered as the initial pivot element.
It is a linear algorithm.	It is also a linear algorithm.
It is relatively faster.	It is slower.
It is slightly difficult to understand and to implement.	It is easy to understand and easy to implement.
It doesn’t fix the pivot element in the correct position.	It fixes the pivot element in the correct position.

Source : https://en.wikipedia.org/wiki/Quicksort#Hoare_partition_scheme
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